User blog:Holomanga/The Axiom of Extensionality
Set theory is the theory of things that can contain other things. This is expressed formally through the magic symbol \in ; x \in A should be read as " x is an element of A ". What makes it of interest is its amazing expressive power - by adding in the appropriate definitions, a large chunk of modern mathematics can be described in terms of sets. As the system in which much of mathematics is formalised, it makes a good place to start, both philosophically and pedagogically. Set theory itself can be summarised in terms of a small collection of facts, from which most of the other facts that one would want to know can be derived. These facts are called axioms, and come in three types: * Definitions of symbols * Sets that exist * Sets that can be made from other sets Since the only new symbol that we've encountered so far is that mysterious \in , let's begin with an axiom that defines that. The Axiom of Extensionality The Axiom of Extensionality gives a definition of \in , which is the thing that distinguishes set theory from boring old formal logic. It is \forall A \forall B \left( \forall z \left( z \in A \leftrightarrow z \in B \right) \rightarrow A = B \right) . This means that if two sets have the same elements, then they are the same set. This tells us what it means for something to be an element of something else; it's the thing that makes two sets equal if everything that does it to one set also does it to the other set. One might wonder what would happen if that biconditional was relaxed to a mere conditional; if all of the elements of A were in B , but not necessarily all of the elements of B are in A . This defines a new symbol, \subseteq , by \left( \forall z \left( z \in x \rightarrow z \in y \right) := x \subseteq y \right) . A \subseteq B should be read as " A is a subset of B ". Note that this definition of \subseteq isn't an axiom, it's just shorthand. You could go through all of set theory only writing the long definition of it, it would just be less convenient. Now that we have some definitions, it's time for the most important part of set theory: proving theorems! Theorems is how the world is done. The big appeal of set theory is really how everything can be done from it with definitions and theorems: you don't need much ad hoc stuff here. These theorems will prove some important properties of the subset relationship: that it is transitive, and that it is antisymmetric. The latter property is often helpful in proofs about whether two sets are equal. Those theorems are pretty basic, but we do only have one axiom to work with, which doesn't even tell us how to make new sets! More axioms will give us more theorems, and therefore more power. Onwards! Category:Blog posts